On counting subring-submodules of free modules over finite commutative Frobenius rings
Let \(R\) be a finite Frobenius commutative ring and \(S\) a finite Galois extension ring of \(R\) of degree \(m\). For each free \(S\)-submodule \(B\) of the free module \(S^{\ell}\) let \(B'=B\cap R^{\ell}\) be the corresponding \(R\)-submodule of \(R^{\ell}\). For positive integers \(k,k'\), the main result of the paper under review is a formula for the number of free \(S\)-submodules \(B\) of the free module \(S^{\ell}\) such that \(S\)-rank of \(B\) is \(k\) and the \(R\)-rank of \(B'\) is \(k'\). The formula is obtained first in the case when \(R\) is a local ring (Theorem 4) and then in the general case (Theorem 6) by decomposing \(R\) as a finite direct product of local Frobenius rings given by the quotiens of \(R\) by adequate powers of the maximal ideals of \(R\). For \({\mathbb F}_q\) a finite field and \({\mathbb F}_{q^m}\) a finite extension of degree \(m\), a by-product of the main result of the paper is a correct formula for the number of distinct \({\mathbb F}_{q^m}\) -linear codes of length \(\ell\), dimension \(k\), and a common fixed subcode over \({\mathbb F}_q\). As the authors point out, this corrects a wrong formula in Theorem 6 of [\textit{B. Lyle}, Linear Algebra Appl. 22, 223--233 (1978; Zbl 0411.15010)].
scientific article; zbMATH DE number 6933688
scientific article